964: For Polynomials Ring over Field, Irreducibles Are Nonconstants That Can Be Factorized Only with Only 1 Nonconstant
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description/proof of that for polynomials ring over field, irreducibles are nonconstants that can be factorized only with only 1 nonconstant
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Starting Context
Target Context
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The reader will have a description and a proof of the proposition that for the polynomials ring over any field, the irreducibles are the nonconstants that can be factorized only with only 1 nonconstant.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
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Statements:
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2: Note
If you are wondering how that Statements amounts to "the nonconstants that can be factorized only with only 1 nonconstant", supposing , if there were 2 nonconstant factors, say and , , and or would not be any constant, a contradiction against the Statements; if there was no constant factor, , and or would not be any constant, a contradiction against the Statements.
Compare with the proposition that for the polynomials ring over any integral domain, any irreducible can be factorized with at most 1 nonconstant.
3: Proof
Whole Strategy: Step 1: let be any irreducible, suppose that and were some nonconstants, find a contradiction; Step 2: let , and see that is irreducible.
Step 1:
Let be any irreducible.
is a nonconstant, because any constant is a unit, by the proposition that for the polynomials ring over any field, the units are the nonzero constants, and any unit is not any irreducible, by the definition of irreducible element of commutative ring.
Let us suppose that .
Let us suppose that and were some nonconstants. or would not be any unit, by the proposition that for the polynomials ring over any field, the units are the nonzero constants. Then, would not be any irreducible, by the definition of irreducible element of commutative ring, a contradiction.
So, or is a constant.
That means that .
By the way, when is a constant, is a nonconstant, because otherwise, would not be nonconstant, a contradiction; when is a constant, is a nonconstant, because otherwise, would not be nonconstant, a contradiction.
Step 2:
Let be any.
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As is a nonconstant, is not any unit, by the proposition that for the polynomials ring over any field, the units are the nonzero constants.
Whenever , is a constant or is a constant. But for the former case, is a unit, by the proposition that for the polynomials ring over any field, the units are the nonzero constants, and for the latter case, is a unit, by the proposition that for the polynomials ring over any field, the units are the nonzero constants.
So, is an irreducible, by the definition of irreducible element of commutative ring.
References
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